{"title":"Symplectic instability of Bézout’s theorem","authors":"","doi":"10.1007/s11856-023-2598-1","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We investigate the failure of Bézout’s Theorem for two symplectic surfaces in ℂP<sup>2</sup> (and more generally on an algebraic surface), by proving that every plane algebraic curve <em>C</em> can be perturbed in the <span> <span>\\({{\\cal C}^\\infty }\\)</span> </span>-topology to an arbitrarily close smooth symplectic surface <em>C</em><sub><em>ϵ</em></sub> with the property that the cardinality #<em>C</em><sub><em>ϵ</em></sub> ∩ <em>Z</em><sub><em>d</em></sub> of the transversal intersection of <em>C</em><sub><em>ϵ</em></sub> with an algebraic plane curve <em>Z</em><sub><em>d</em></sub> of degree <em>d</em>, as a function of <em>d</em>, can grow arbitrarily fast. As a consequence we obtain that, although Bézout’s Theorem is true for pseudoholomorphic curves with respect to the same almost complex structure, it is “arbitrarly false” for pseudoholomorphic curves with respect to different (but arbitrarily close) almost-complex structures (we call this phenomenon “instability of Bézout’s Theorem”).</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Israel Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11856-023-2598-1","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
We investigate the failure of Bézout’s Theorem for two symplectic surfaces in ℂP2 (and more generally on an algebraic surface), by proving that every plane algebraic curve C can be perturbed in the \({{\cal C}^\infty }\)-topology to an arbitrarily close smooth symplectic surface Cϵ with the property that the cardinality #Cϵ ∩ Zd of the transversal intersection of Cϵ with an algebraic plane curve Zd of degree d, as a function of d, can grow arbitrarily fast. As a consequence we obtain that, although Bézout’s Theorem is true for pseudoholomorphic curves with respect to the same almost complex structure, it is “arbitrarly false” for pseudoholomorphic curves with respect to different (but arbitrarily close) almost-complex structures (we call this phenomenon “instability of Bézout’s Theorem”).
期刊介绍：
The Israel Journal of Mathematics is an international journal publishing high-quality original research papers in a wide spectrum of pure and applied mathematics. The prestigious interdisciplinary editorial board reflects the diversity of subjects covered in this journal, including set theory, model theory, algebra, group theory, number theory, analysis, functional analysis, ergodic theory, algebraic topology, geometry, combinatorics, theoretical computer science, mathematical physics, and applied mathematics.
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